Documentation

Mathlib.Algebra.Module.ZMod

The ZMod n-module structure on Abelian groups whose elements have order dividing n #

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@[reducible, inline]
abbrev AddCommMonoid.zmodModule {n : ℕ} {M : Type u_1} [NeZero n] [AddCommMonoid M] (h : ∀ (x : M), n • x = 0) :
Module (ZMod n) M

The ZMod n-module structure on commutative monoids whose elements have order dividing n ≠ 0. Also implies a group structure via Module.addCommMonoidToAddCommGroup. See note [reducible non-instances].

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      @[reducible, inline]
      abbrev AddCommGroup.zmodModule {n : ℕ} {G : Type u_3} [AddCommGroup G] (h : ∀ (x : G), n • x = 0) :
      Module (ZMod n) G

      The ZMod n-module structure on Abelian groups whose elements have order dividing n. See note [reducible non-instances].

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          @[reducible, inline]
          abbrev QuotientAddGroup.zmodModule {n : ℕ} {G : Type u_3} [AddCommGroup G] {H : AddSubgroup G} (hH : ∀ (x : G), n • x ∈ H) :
          Module (ZMod n) (G ⧸ H)

          The quotient of an abelian group by a subgroup containing all multiples of n is a n-torsion group.

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              theorem ZMod.map_smul {n : ℕ} {M : Type u_1} {M₁ : Type u_2} {F : Type u_3} [AddCommGroup M] [AddCommGroup M₁] [FunLike F M M₁] [AddMonoidHomClass F M M₁] [Module (ZMod n) M] [Module (ZMod n) M₁] (f : F) (c : ZMod n) (x : M) :
              f (c • x) = c • f x
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              theorem ZMod.smul_mem {n : ℕ} {M : Type u_1} {S : Type u_4} [AddCommGroup M] [Module (ZMod n) M] [SetLike S M] [AddSubgroupClass S M] {x : M} {K : S} (hx : x ∈ K) (c : ZMod n) :
              c • x ∈ K
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              def AddMonoidHom.toZModLinearMap (n : ℕ) {M : Type u_1} {M₁ : Type u_2} [AddCommGroup M] [AddCommGroup M₁] [Module (ZMod n) M] [Module (ZMod n) M₁] (f : M →+ M₁) :
              M →ₗ[ZMod n] M₁

              Reinterpret an additive homomorphism as a ℤ/nℤ-linear map.

              See also: AddMonoidHom.toIntLinearMap, AddMonoidHom.toNatLinearMap, AddMonoidHom.toRatLinearMap

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                  theorem AddMonoidHom.toZModLinearMap_injective (n : ℕ) {M : Type u_1} {M₁ : Type u_2} [AddCommGroup M] [AddCommGroup M₁] [Module (ZMod n) M] [Module (ZMod n) M₁] :
                  Function.Injective (toZModLinearMap n)
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                  @[simp]
                  theorem AddMonoidHom.coe_toZModLinearMap (n : ℕ) {M : Type u_1} {M₁ : Type u_2} [AddCommGroup M] [AddCommGroup M₁] [Module (ZMod n) M] [Module (ZMod n) M₁] (f : M →+ M₁) :
                  ⇑(toZModLinearMap n f) = ⇑f
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                  def AddMonoidHom.toZModLinearMapEquiv (n : ℕ) {M : Type u_1} {M₁ : Type u_2} [AddCommGroup M] [AddCommGroup M₁] [Module (ZMod n) M] [Module (ZMod n) M₁] :
                  (M →+ M₁) ≃+ (M →ₗ[ZMod n] M₁)

                  AddMonoidHom.toZModLinearMap as an equivalence.

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                      def AddSubgroup.toZModSubmodule (n : ℕ) {M : Type u_1} [AddCommGroup M] [Module (ZMod n) M] :
                      AddSubgroup M ≃o Submodule (ZMod n) M

                      Reinterpret an additive subgroup of a ℤ/nℤ-module as a ℤ/nℤ-submodule.

                      See also: AddSubgroup.toIntSubmodule, AddSubmonoid.toNatSubmodule.

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                          @[simp]
                          theorem AddSubgroup.toZModSubmodule_symm (n : ℕ) {M : Type u_1} [AddCommGroup M] [Module (ZMod n) M] :
                          ⇑(toZModSubmodule n).symm = Submodule.toAddSubgroup
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                          @[simp]
                          theorem AddSubgroup.coe_toZModSubmodule (n : ℕ) {M : Type u_1} [AddCommGroup M] [Module (ZMod n) M] (S : AddSubgroup M) :
                          ↑((toZModSubmodule n) S) = ↑S
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                          @[simp]
                          theorem AddSubgroup.mem_toZModSubmodule (n : ℕ) {M : Type u_1} [AddCommGroup M] [Module (ZMod n) M] {x : M} {S : AddSubgroup M} :
                          x ∈ (toZModSubmodule n) S ↔ x ∈ S
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                          @[simp]
                          theorem AddSubgroup.toZModSubmodule_toAddSubgroup (n : ℕ) {M : Type u_1} [AddCommGroup M] [Module (ZMod n) M] (S : AddSubgroup M) :
                          ((toZModSubmodule n) S).toAddSubgroup = S
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                          @[simp]
                          theorem Submodule.toAddSubgroup_toZModSubmodule (n : ℕ) {M : Type u_1} [AddCommGroup M] [Module (ZMod n) M] (S : Submodule (ZMod n) M) :
                          (AddSubgroup.toZModSubmodule n) S.toAddSubgroup = S
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                          theorem ZModModule.exists_submodule_subset_card_le {p : ℕ} {G : Type u_5} [AddCommGroup G] (hp : Nat.Prime p) [Module (ZMod p) G] (H : Submodule (ZMod p) G) {k : ℕ} (hk : k ≤ Nat.card ↥H) (h'k : k ≠ 0) :
                          ∃ (H' : Submodule (ZMod p) G), Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥H' ∧ H' ≤ H

                          In an elementary abelian p-group, every finite subgroup H contains a further subgroup of cardinality between k and p * k, if k ≤ |H|.